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If you want to make backups and have a USB stick or other device as a second storage device, then you can go to System Settings, Memory, select the device with the saves (hard drive likely) on it, then go to Games. There will be a listing of all the games you have saves for. Underneath each game there could be different files, saves, other items; each game does things a little differently. Sometimes saves are all in one file, sometimes they are split into different files. Using the menu options, you can select game folders or files and select to Copy them or Move them to a different device. In this case, you would want to Copy them. You can read some more about copying at Xbox Support.

As a side note, if your Xbox really gives you a RROD, your game saves should be in a fine state unless it happened to do so right when a save was taking place. A RROD will have no effect on your hard drive or storage device. If you have to ship in your Xbox for repair, you won't ship it in with the hard drive or any storage devices attached so your saves will be fine when you place it on the replacement box they ship to you.

If I get a RROD on my arcade 360 (arcade=internal flash!), then I'll need to send my unit and pray that they don't wipe out contents. But I guess that I can safely backup on a USB stick.
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michelemarconDec 29 '10 at 11:04

@michelemarcon you're correct, hadn't considered the arcade unit. I never got the same machine back when sending in a RROD unit (3 total), so not sure if they transfer memory contents or not as I had one w/ a hard drive.
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Doozer BlakeDec 29 '10 at 13:39